SOME PROPERTIES OF PRIMITIVE HENSTOCK OF INTEGRABLE FUNCTION IN LOCALLY COMPACT METRIC SPACE OF VECTOR VALUED FUNCTION.
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
M-10
SOME PROPERTIES OF PRIMITIVE
HENSTOCK OF INTEGRABLE
FUNCTION IN LOCALLY COMPACT METRIC SPACE OF VECTOR
VALUED FUNCTION
Manuharawati
e-mail: [email protected],
Mathematics Department
Faculty of Mathematics and Natural Science-The State University of Surabaya
Abstract
By abstracting the domain and range of Henstock integrable function and
measures (the volume function) used by mathematicians, we had constructed
Henstock integral in locally compact metric space of vector valued function.
Based on the result of previous researchers, this paper discuss properties of
primitve
Henstock integrable function in locally compact metric space of
vector valued function.
Key words:
vector.
Henstock integrable function , locally compact metric space,
1. INTRODUCTION
1.1 Background
In 1989, Lee had contructed Henstock integral of real valued fungtion on a cell
.
Manuharawati (2002) generalized this integral by reduced a cell
to a cell in a
locally compact metric space and Cao (1992) generalized this integral by reduced the range of
function from to Banach space.
Based on the result of these researchers, Manuharawati and Yunianti (2013) had contructed
a Henstock integral of vector valued function (the range of function is a Banach space ) in
locally compact metric space. In this integral, we used a volume function that defined on the
set of all elementary sets in a locally compact metric space. In this paper, we discuss some
properties of the primitive this integral.
1.2 Fundamental Concepts
Let be a locally compact metric space,
be an extended real numbers,
real numbers, and
be a collection of elementary sets in .
be a positive
A nonvoid collection S 2X is called a system of intervals if it satisfies: (Manuharawati,
2002: 11-12):
(i) for every p X, {p} S,
(ii) for every p X, if cl(N(p,r)) compact then N(p,r) S ,
(iii) if A S then A konvex, cl(A) compact, cl(A) S, and int(A) S,
M-71
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
(iv) for every A, B S, A B S.
(v) for every A, B S there exists a collection of nonoverlapping sets Ci S,
that
such
Every an element of S is called an interval. Interval A S is said to be degenerate if
. A cell is a nondegenerate compact interval. A
and nondegenerate if
set A X is called elementary set if A is a finite union of intervals.
A Volume function
(i)
(ii)
is called a measure on
if
and
(B) for every A, B
if:
if
with A B,
for every nonoverlapping collection sets {Ai} E(X) with
(iii)
.
In this paper
is a nondiscrete locally compact metric space.
If E is a cell and
, then a finite collection of cell-point pairs
=
is called a Perron -fine partition on E if
on E.
If
is a cell and
(Manuharawati, 2002: 18).
, then there exists a Perron
A real function
on a vector space
and
satisfies:
(i)
(ii)
(iii)
(iv)
and
(over field
is a partition
-fine partition on E
) is called a norm on
if for every
,
A vector space
norm space.
,
,
.
thegether a norm is called a norm space. A Banach space is a complete
A function
is said to be Henstock - integrable on a cell E X there exists a
such that
vector
such that for every real number 0 thre exists a function
for every Perron -fine partition
=
on E we have
=
If A S with
, then integral- Henstock of a function
on A is defined by
null vectorl (O). The set of all
Henstock integrable function on a cell
is denoted by
P-72
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
and a vector a is called
Henstock integral value of a function f on E and denoted
by
(Manuharawati and Yunianti, 2013: 20-21).
Some properties of the
discussion are as follow.
Henstock integrable function which will be used in
Theorem 1.2.1: Cauchy’s Criterian (Manuharawati and Yunianti, 2013a: 71-75): A functin
such
if and only if for any real number 0 there exists a function
that for any Perron -fine partition
and
on E we have
.
Theorem 1.2.2: (Manuharawati and Yunianti, 2013: 24) If
,
.
, Then for every a cell
Theorem 1.2.3: (Manuharawati, 2013: 22-26) A function
is Henstock - integrable
on a cell E X if only if there exists an additive function
with property for any real
such that for any Perron -fine partition
number 0 there exists a function
=
on ,
.
If
,
is a cell and
is a collection of all subinterval in , then an additive function
is called
Henstock primitive of on if
(null vektor) and for every
, we have
.
M-73
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
Henstock
Theorem 1.2.4: Henstock’s Lemma (Manuharawati, 2013: 22-26): If
integrable on a cell E X with primitive , then for any real number 0 there exists a
shuch that for any Perron -fine partition
and any
function
we have
.
2.
RESULT AND DISCUSSION
Let be a nondicret metric space, be Banach space,
be a volume function on
, and be a cell in . We start define an interval continuous function relative to volume
function .
Definition 2.1: A function
for every real number
with
,
is said to be
continuous in an interval
there exist a real number
such that for every
we have
and
A function G is said to be
.
.
continuous on
Lemma 2.2: If an additive function
number
there exist a real number
we have
if
if G is
continuous in every interval
is
continuous on
then for every real
such that for every interval
with
.
Proof: Let
for every
interval
be given. Since is
continuous on
, then is
continuous at
. So for every
there exist a real number
such that for every open
with
and
we
.
If
,
then
.
is a vitali cover of . Since
with
is compact set, then we have
is an open cover of .
Further, put
P-74
such that
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
.
Since
is an additive function, then for every interval
with
we have
.
Definition 2.3: A function
if for every real number
in with
partition
is said to be strongly absolutely
continuous on
there exist a real number
such that for every
and
we have
.
Theorem 2.4: Let a function
on is and is bounded, then
Proof: From boundednes of
implies
be given. If
is trongly absolutly
with
Henstock primitve of
continuous on .
on , there exist a real number
such that for every
.
Given any real number
. Based on density property on , there exists a real number
. Take a collection
,
, with
such that
.
Since
, then by Henstock’s Lemma and Theorem 1.2.3, there exist a function
on
we have
such that for every
fine partition
Since
is a
Henstock primitive of
on , then by Theorem 1.2.2 for every
we have
+
.
So we have
This mean that
= .
is strongly absolutly
continuous on .
Theorem 2.5: Let a function
be given. If
on is , then is
continuous on
.
Proof: Let
such that
, then
with
Henstock primitve of
be areal positive number. If is a cell in
, then there exists open set
in
and
. If be a collection of all
with
for all
is an open cover of . Since is a compact set, then there exists
M-75
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
such that
If
if
, then by Lemma 2.2 and additivity of , there exists areal number
we have
.
such that
Let
and
be cells with
,
and
Then there exist two finite collection of nonoverlapping cells
and
such that
and
Since
and
, then for eny
is an additive function on
. So, we have
and
So, is
continuous on
.
Theorem 2.6: Let a function
be given. If
on
is , then
exists allmost everywhere on
everywhere on .
Proof: Given a real number
. Since
by Henstock’s Lemma, there exist a function
partition
on and every subpartition
.
.
and
. Note that,
.
with
and
Henstock primitve of
allmost
with
Henstock primitive is
such that for every Perron
of we have
.
, then
fine
If
then for every
there exists
,
with property: for every real number
there exists a real number
shuch that
.
For every
, put
.
So wehave
.
If
then
is is vitali cover for
,
. So, there exists a nonoverlapping collection of cell
such that
.
This mean that
and then
or
P-76
exists allmost everywhere on .
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
If
then we define shape of
as follow
.
Further, if
and
be given, a collection of all cell
, denoted by
.
with property
Theorem 2.7: Given a function
. If
with
Henstock primitve of on
is , then for every real number
there exist a real number
such that for every
interval
with
we have
.
Proof: Since is a
Henstock primitve of on , then is an additive function on
Lemma 2.2 and Theorem 2.5, then for every real number
there exist a real number
such that for every interval
with
we have
. By
.
3. CONCLUSION
If is a locally compact metric space, is a Banach space, and is a
Henstock primitive of
a function
on a cell , then we have:
3.1
is -continuous on
,
3.2
is strongly absolutely -continuous on
if is a bounded on ,
3.3
exists allmost everywhere on and
allmost everywhere on ,
3.4 for every real number
there exist a real number
such that for every interval
with
we have
.
REFERENCES
Cao,S.S., 1992. The Henstock Integral for Banach Valued Function. SEA Bull. Math..16,1:
35 – 40.
Lee, P. Y., 1989, Lanzhou Lectures on Hensock Integration. World Scientific Publishing
Co.Pte.Ltd. Singapore.
Manuharawati. 2002. Integral Henstock di dalam Ruang Metrik Kompak Lokal. Disertasi S3.
PPs-UGM, Yogyakarta.
Manuharawati, 2013. Lemma Henstock Untuk Suatu Fungsi Bernilai Vektor di dalam Ruang
Metrik Kompak Lokal. Jurnal Sains & Matematika. Vol. 2, No: 1. FMIPA Unesa
Surabaya.
M-77
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
Manuharawati dan Yunianti, D. N. 2013a. Kriteria Cauchy dari suatu Fungsi Bernilai Vektor
Pada Suatu Sel Di Dalam Ruang metrik Kompak lokal. Prosiding Seminar Nasional
Matematika dan Aplikasinya, 2013. Unair Surabaya.
Manuharawati dan Yunianti, D. N. 2013. Integral
Henstock di dalam Ruang Metrik Kompak
Lokal Dari Fungsi Bernilai Vektor. Laporan Penelitian. LPPM Unesa. Surabaya.
P-78
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
M-10
SOME PROPERTIES OF PRIMITIVE
HENSTOCK OF INTEGRABLE
FUNCTION IN LOCALLY COMPACT METRIC SPACE OF VECTOR
VALUED FUNCTION
Manuharawati
e-mail: [email protected],
Mathematics Department
Faculty of Mathematics and Natural Science-The State University of Surabaya
Abstract
By abstracting the domain and range of Henstock integrable function and
measures (the volume function) used by mathematicians, we had constructed
Henstock integral in locally compact metric space of vector valued function.
Based on the result of previous researchers, this paper discuss properties of
primitve
Henstock integrable function in locally compact metric space of
vector valued function.
Key words:
vector.
Henstock integrable function , locally compact metric space,
1. INTRODUCTION
1.1 Background
In 1989, Lee had contructed Henstock integral of real valued fungtion on a cell
.
Manuharawati (2002) generalized this integral by reduced a cell
to a cell in a
locally compact metric space and Cao (1992) generalized this integral by reduced the range of
function from to Banach space.
Based on the result of these researchers, Manuharawati and Yunianti (2013) had contructed
a Henstock integral of vector valued function (the range of function is a Banach space ) in
locally compact metric space. In this integral, we used a volume function that defined on the
set of all elementary sets in a locally compact metric space. In this paper, we discuss some
properties of the primitive this integral.
1.2 Fundamental Concepts
Let be a locally compact metric space,
be an extended real numbers,
real numbers, and
be a collection of elementary sets in .
be a positive
A nonvoid collection S 2X is called a system of intervals if it satisfies: (Manuharawati,
2002: 11-12):
(i) for every p X, {p} S,
(ii) for every p X, if cl(N(p,r)) compact then N(p,r) S ,
(iii) if A S then A konvex, cl(A) compact, cl(A) S, and int(A) S,
M-71
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
(iv) for every A, B S, A B S.
(v) for every A, B S there exists a collection of nonoverlapping sets Ci S,
that
such
Every an element of S is called an interval. Interval A S is said to be degenerate if
. A cell is a nondegenerate compact interval. A
and nondegenerate if
set A X is called elementary set if A is a finite union of intervals.
A Volume function
(i)
(ii)
is called a measure on
if
and
(B) for every A, B
if:
if
with A B,
for every nonoverlapping collection sets {Ai} E(X) with
(iii)
.
In this paper
is a nondiscrete locally compact metric space.
If E is a cell and
, then a finite collection of cell-point pairs
=
is called a Perron -fine partition on E if
on E.
If
is a cell and
(Manuharawati, 2002: 18).
, then there exists a Perron
A real function
on a vector space
and
satisfies:
(i)
(ii)
(iii)
(iv)
and
(over field
is a partition
-fine partition on E
) is called a norm on
if for every
,
A vector space
norm space.
,
,
.
thegether a norm is called a norm space. A Banach space is a complete
A function
is said to be Henstock - integrable on a cell E X there exists a
such that
vector
such that for every real number 0 thre exists a function
for every Perron -fine partition
=
on E we have
=
If A S with
, then integral- Henstock of a function
on A is defined by
null vectorl (O). The set of all
Henstock integrable function on a cell
is denoted by
P-72
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
and a vector a is called
Henstock integral value of a function f on E and denoted
by
(Manuharawati and Yunianti, 2013: 20-21).
Some properties of the
discussion are as follow.
Henstock integrable function which will be used in
Theorem 1.2.1: Cauchy’s Criterian (Manuharawati and Yunianti, 2013a: 71-75): A functin
such
if and only if for any real number 0 there exists a function
that for any Perron -fine partition
and
on E we have
.
Theorem 1.2.2: (Manuharawati and Yunianti, 2013: 24) If
,
.
, Then for every a cell
Theorem 1.2.3: (Manuharawati, 2013: 22-26) A function
is Henstock - integrable
on a cell E X if only if there exists an additive function
with property for any real
such that for any Perron -fine partition
number 0 there exists a function
=
on ,
.
If
,
is a cell and
is a collection of all subinterval in , then an additive function
is called
Henstock primitive of on if
(null vektor) and for every
, we have
.
M-73
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
Henstock
Theorem 1.2.4: Henstock’s Lemma (Manuharawati, 2013: 22-26): If
integrable on a cell E X with primitive , then for any real number 0 there exists a
shuch that for any Perron -fine partition
and any
function
we have
.
2.
RESULT AND DISCUSSION
Let be a nondicret metric space, be Banach space,
be a volume function on
, and be a cell in . We start define an interval continuous function relative to volume
function .
Definition 2.1: A function
for every real number
with
,
is said to be
continuous in an interval
there exist a real number
such that for every
we have
and
A function G is said to be
.
.
continuous on
Lemma 2.2: If an additive function
number
there exist a real number
we have
if
if G is
continuous in every interval
is
continuous on
then for every real
such that for every interval
with
.
Proof: Let
for every
interval
be given. Since is
continuous on
, then is
continuous at
. So for every
there exist a real number
such that for every open
with
and
we
.
If
,
then
.
is a vitali cover of . Since
with
is compact set, then we have
is an open cover of .
Further, put
P-74
such that
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
.
Since
is an additive function, then for every interval
with
we have
.
Definition 2.3: A function
if for every real number
in with
partition
is said to be strongly absolutely
continuous on
there exist a real number
such that for every
and
we have
.
Theorem 2.4: Let a function
on is and is bounded, then
Proof: From boundednes of
implies
be given. If
is trongly absolutly
with
Henstock primitve of
continuous on .
on , there exist a real number
such that for every
.
Given any real number
. Based on density property on , there exists a real number
. Take a collection
,
, with
such that
.
Since
, then by Henstock’s Lemma and Theorem 1.2.3, there exist a function
on
we have
such that for every
fine partition
Since
is a
Henstock primitive of
on , then by Theorem 1.2.2 for every
we have
+
.
So we have
This mean that
= .
is strongly absolutly
continuous on .
Theorem 2.5: Let a function
be given. If
on is , then is
continuous on
.
Proof: Let
such that
, then
with
Henstock primitve of
be areal positive number. If is a cell in
, then there exists open set
in
and
. If be a collection of all
with
for all
is an open cover of . Since is a compact set, then there exists
M-75
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
such that
If
if
, then by Lemma 2.2 and additivity of , there exists areal number
we have
.
such that
Let
and
be cells with
,
and
Then there exist two finite collection of nonoverlapping cells
and
such that
and
Since
and
, then for eny
is an additive function on
. So, we have
and
So, is
continuous on
.
Theorem 2.6: Let a function
be given. If
on
is , then
exists allmost everywhere on
everywhere on .
Proof: Given a real number
. Since
by Henstock’s Lemma, there exist a function
partition
on and every subpartition
.
.
and
. Note that,
.
with
and
Henstock primitve of
allmost
with
Henstock primitive is
such that for every Perron
of we have
.
, then
fine
If
then for every
there exists
,
with property: for every real number
there exists a real number
shuch that
.
For every
, put
.
So wehave
.
If
then
is is vitali cover for
,
. So, there exists a nonoverlapping collection of cell
such that
.
This mean that
and then
or
P-76
exists allmost everywhere on .
Proceeding of International Conference On Research, Implementation And Education
Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
If
then we define shape of
as follow
.
Further, if
and
be given, a collection of all cell
, denoted by
.
with property
Theorem 2.7: Given a function
. If
with
Henstock primitve of on
is , then for every real number
there exist a real number
such that for every
interval
with
we have
.
Proof: Since is a
Henstock primitve of on , then is an additive function on
Lemma 2.2 and Theorem 2.5, then for every real number
there exist a real number
such that for every interval
with
we have
. By
.
3. CONCLUSION
If is a locally compact metric space, is a Banach space, and is a
Henstock primitive of
a function
on a cell , then we have:
3.1
is -continuous on
,
3.2
is strongly absolutely -continuous on
if is a bounded on ,
3.3
exists allmost everywhere on and
allmost everywhere on ,
3.4 for every real number
there exist a real number
such that for every interval
with
we have
.
REFERENCES
Cao,S.S., 1992. The Henstock Integral for Banach Valued Function. SEA Bull. Math..16,1:
35 – 40.
Lee, P. Y., 1989, Lanzhou Lectures on Hensock Integration. World Scientific Publishing
Co.Pte.Ltd. Singapore.
Manuharawati. 2002. Integral Henstock di dalam Ruang Metrik Kompak Lokal. Disertasi S3.
PPs-UGM, Yogyakarta.
Manuharawati, 2013. Lemma Henstock Untuk Suatu Fungsi Bernilai Vektor di dalam Ruang
Metrik Kompak Lokal. Jurnal Sains & Matematika. Vol. 2, No: 1. FMIPA Unesa
Surabaya.
M-77
Manuharawati / Some Properties of....
ISBN. 978-979-99314-8-1
Manuharawati dan Yunianti, D. N. 2013a. Kriteria Cauchy dari suatu Fungsi Bernilai Vektor
Pada Suatu Sel Di Dalam Ruang metrik Kompak lokal. Prosiding Seminar Nasional
Matematika dan Aplikasinya, 2013. Unair Surabaya.
Manuharawati dan Yunianti, D. N. 2013. Integral
Henstock di dalam Ruang Metrik Kompak
Lokal Dari Fungsi Bernilai Vektor. Laporan Penelitian. LPPM Unesa. Surabaya.
P-78